Betting-Time Analysis (Normal Condition Only)
analysis done by Henrik Singmann (singmann@gmail.com)
01 March, 2022
Preparation
Total participants (NULL = did not pass captchas):
##
## delay normal NULL
## 502 502 182Participants that passed the captchas and finished the study:
##
## Normal Slowed-down
## 500 502All following analysis only use the normal (i.e., not slowed down) condition.
Distribution of Betting Times
We considered the time participants took for each round of roulette starting with the second round (as this was the first round to which the 1 minute slow down applied).
The first graph shows the distribution of individual betting times in seconds when restricting the shown range to up to 150 seconds. The first vertical line shows the overall median and the second line the overall mean (which as expected is strongly affected by the outliers on the right side).
This plot excludes a few very long betting times.
## # A tibble: 2 × 3
## ppt_id exp_cond time
## <chr> <fct> <dbl>
## 1 P0899 Normal 433
## 2 P0787 Normal 888We can also look at some overall statistics of the betting times:
## # A tibble: 1 × 5
## exp_cond time_mean time_median time_sd time_IQR
## <fct> <dbl> <dbl> <dbl> <dbl>
## 1 Normal 20.95 15 25.73 10Let’s take a look how the mean changes when using certain exclusion cut-offs (co):
## # A tibble: 1 × 9
## exp_cond m_co_150 m_co_120 m_co_100 m_co_90 m_co_80 m_co_75 m_co_60 m_co_50
## <fct> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 Normal 19.77 19.58 19.41 19.26 19.15 18.97 18.37 17.65These exclusion criteria would lead to the following proportions of excluded observations:
## # A tibble: 8 × 3
## exp_cond cut_off prop_excluded
## <fct> <chr> <dbl>
## 1 Normal prop_co_150 0.00466
## 2 Normal prop_co_120 0.00621
## 3 Normal prop_co_100 0.00777
## 4 Normal prop_co_90 0.00971
## 5 Normal prop_co_80 0.0117
## 6 Normal prop_co_75 0.0136
## 7 Normal prop_co_60 0.0260
## 8 Normal prop_co_50 0.0443Distribution of mean betting times across participants
In the following output and plot, the value for excl (also given in the panel header) gives the cut-off for individual response times.
## # A tibble: 9 × 4
## excl mean median sd
## <dbl> <dbl> <dbl> <dbl>
## 1 50 19.7 17.4 7.34
## 2 60 21.1 18.1 9.34
## 3 75 21.9 18.5 10.6
## 4 80 22.0 18.5 10.8
## 5 90 22.2 18.5 10.9
## 6 100 22.3 18.5 11.1
## 7 120 22.5 18.5 11.9
## 8 150 22.5 18.7 11.9
## 9 Inf 23.2 19 13.2
Modelling mean betting times (no cut-off)
Normal Model
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: time ~ 1 + (1 | ppt_id)
## Data: bet_times (Number of observations: 2575)
## Draws: 6 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup draws = 6000
##
## Group-Level Effects:
## ~ppt_id (Number of levels: 318)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sd(Intercept) 8.09 0.75 6.66 9.57 1.00 2721 4176
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 22.06 0.76 20.56 23.52 1.00 4328 4232
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 24.45 0.35 23.77 25.16 1.00 8988 4555
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Lognormal Model
## Family: lognormal
## Links: mu = identity; sigma = identity
## Formula: time ~ 1 + (1 | ppt_id)
## Data: bet_times (Number of observations: 2575)
## Draws: 6 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup draws = 6000
##
## Group-Level Effects:
## ~ppt_id (Number of levels: 318)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sd(Intercept) 0.35 0.02 0.31 0.39 1.00 1289 2284
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 2.95 0.02 2.91 3.00 1.01 1023 1895
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.39 0.01 0.38 0.41 1.00 7343 4324
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Model quantities and Statistics
Minimum and maximum betting times in the data:
## [1] 9## [1] 888Minimum and maximum in 11 draws from the posterior predictive distribution:
## [1] 4.039240 3.564740 3.852988 3.278238 4.475241 3.570153 3.910283 2.927886 3.662925 3.659125 2.611115## [1] 125.0516 101.4812 104.7116 132.4454 114.4747 153.7466 103.6929 101.1210 122.0891 111.9899 163.4144Let’s take a look at posterior predicted mean and median
## [1] 19.86070 20.26657 19.94629 19.81310 20.07192 19.68474 19.85559 19.79141 20.17550 19.68163 19.73571## [1] 16.84672 17.04487 17.38620 17.09183 17.26422 16.70779 17.18313 17.22558 17.52457 16.97000 16.73180Calculate predicted mean and predicted median (based on lognormal distribution)
## # A tibble: 1 × 6
## mean .lower .upper .width .point .interval
## <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 20.7 19.8 21.7 0.95 mean hdi## # A tibble: 1 × 6
## median .lower .upper .width .point .interval
## <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 19.2 18.3 20.1 0.95 mean hdiShifted Lognormal Model
## Family: shifted_lognormal
## Links: mu = identity; sigma = identity; ndt = identity
## Formula: time ~ 1 + (1 | ppt_id)
## Data: bet_times (Number of observations: 2575)
## Draws: 6 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup draws = 6000
##
## Group-Level Effects:
## ~ppt_id (Number of levels: 318)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sd(Intercept) 0.70 0.04 0.63 0.77 1.00 1268 2413
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 2.13 0.05 2.04 2.22 1.01 882 1678
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.76 0.01 0.74 0.79 1.00 6173 5179
## ndt 8.73 0.04 8.65 8.79 1.00 6983 5206
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Model quantities and Statistics
Minimum and maximum betting times in the data:
## [1] 9## [1] 888Minimum and maximum in 11 draws from the posterior predictive distribution:
## [1] 9.174398 8.979860 8.868114 8.898929 8.993298 8.848665 8.863397 8.998706 8.811846 8.814202 9.047850## [1] 258.1951 289.5646 347.4561 171.9492 288.9181 255.8986 433.0832 251.9624 230.9003 212.0426 201.8293Let’s take a look at posterior predicted mean and median
## [1] 20.65065 20.45311 20.15236 19.74793 20.48158 20.49463 20.62867 20.19683 20.38539 19.94695 19.90674## [1] 15.54920 15.68571 15.20124 15.20125 15.47504 15.91139 15.94550 15.39801 15.51341 15.25097 15.57314Calculate predicted mean and predicted median (based on shifted-lognormal distribution)
## # A tibble: 1 × 6
## mean .lower .upper .width .point .interval
## <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 20.0 19.0 21.0 0.95 mean hdi## # A tibble: 1 × 6
## median .lower .upper .width .point .interval
## <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 17.1 16.3 17.9 0.95 mean hdiModelling mean betting times (cut-off: 90 seconds)
Normal Model
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: time ~ 1 + (1 | ppt_id)
## Data: bet_times_90 (Number of observations: 2549)
## Draws: 6 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup draws = 6000
##
## Group-Level Effects:
## ~ppt_id (Number of levels: 318)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sd(Intercept) 8.67 0.47 7.78 9.66 1.01 1437 2647
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 21.54 0.57 20.45 22.64 1.00 811 2117
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 8.92 0.14 8.66 9.20 1.00 7228 4324
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Lognormal Model
## Family: lognormal
## Links: mu = identity; sigma = identity
## Formula: time ~ 1 + (1 | ppt_id)
## Data: bet_times_90 (Number of observations: 2549)
## Draws: 6 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup draws = 6000
##
## Group-Level Effects:
## ~ppt_id (Number of levels: 318)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sd(Intercept) 0.35 0.02 0.31 0.38 1.01 919 2079
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 2.94 0.02 2.90 2.98 1.01 666 1229
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.33 0.00 0.33 0.34 1.00 6971 4636
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Model quantities and Statistics
Minimum and maximum betting times in the data:
## [1] 9## [1] 86Minimum and maximum in 11 draws from the posterior predictive distribution:
## [1] 5.226036 4.162279 4.047635 4.471974 4.412363 4.578881 3.209284 4.392020 4.122834 3.792337 4.256722## [1] 91.12562 88.13151 100.23247 86.33147 94.51696 74.14926 102.14024 158.22418 93.69908 94.20880
## [11] 111.93499Let’s take a look at posterior predicted mean and median
## [1] 18.91706 18.91130 19.13732 19.10190 19.02086 18.91457 19.08634 18.99226 19.09132 18.81528 19.29700## [1] 16.75222 16.90527 16.59575 16.84519 16.78974 16.59222 16.79071 16.76565 16.73795 16.77850 16.81987Calculate predicted mean and predicted median (based on lognormal distribution)
## # A tibble: 1 × 6
## mean .lower .upper .width .point .interval
## <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 20.0 19.2 20.9 0.95 mean hdi## # A tibble: 1 × 6
## median .lower .upper .width .point .interval
## <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 18.9 18.1 19.7 0.95 mean hdiShifted Lognormal Model
## Family: shifted_lognormal
## Links: mu = identity; sigma = identity; ndt = identity
## Formula: time ~ 1 + (1 | ppt_id)
## Data: bet_times_90 (Number of observations: 2549)
## Draws: 6 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup draws = 6000
##
## Group-Level Effects:
## ~ppt_id (Number of levels: 318)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sd(Intercept) 0.68 0.04 0.61 0.75 1.01 972 2321
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 2.13 0.04 2.04 2.22 1.00 937 1766
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.69 0.01 0.67 0.72 1.00 5642 4494
## ndt 8.62 0.05 8.52 8.71 1.00 6108 4718
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Model quantities and Statistics
Minimum and maximum betting times in the data:
## [1] 9## [1] 86Minimum and maximum in 11 draws from the posterior predictive distribution:
## [1] 9.095918 8.831445 8.863372 8.864292 8.906503 8.929909 8.919281 8.985343 8.849019 8.983668 8.879362## [1] 343.3229 177.8924 186.5252 184.4259 320.1171 221.6220 199.7665 145.7297 219.1756 115.4505 168.9572Let’s take a look at posterior predicted mean and median
## [1] 19.36524 19.15152 19.54800 19.21169 19.60921 19.65871 19.78737 18.87787 19.57208 19.18441 19.83708## [1] 15.09133 15.19017 15.27785 14.99656 15.39796 15.23725 15.39437 14.75839 15.02865 15.45275 15.45886Calculate predicted mean and predicted median (based on shifted-lognormal distribution)
## # A tibble: 1 × 6
## mean .lower .upper .width .point .interval
## <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 19.3 18.5 20.3 0.95 mean hdi## # A tibble: 1 × 6
## median .lower .upper .width .point .interval
## <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 17.1 16.3 17.8 0.95 mean hdiModelling mean betting times (cut-off: 360 seconds)
Normal Model
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: time ~ 1 + (1 | ppt_id)
## Data: bet_times_360 (Number of observations: 2573)
## Draws: 6 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup draws = 6000
##
## Group-Level Effects:
## ~ppt_id (Number of levels: 318)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sd(Intercept) 8.15 0.59 7.03 9.34 1.00 2075 3264
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 22.00 0.64 20.75 23.26 1.00 2333 3177
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 15.64 0.23 15.20 16.11 1.00 8439 4900
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Lognormal Model
## Family: lognormal
## Links: mu = identity; sigma = identity
## Formula: time ~ 1 + (1 | ppt_id)
## Data: bet_times_360 (Number of observations: 2573)
## Draws: 6 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup draws = 6000
##
## Group-Level Effects:
## ~ppt_id (Number of levels: 318)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sd(Intercept) 0.35 0.02 0.31 0.39 1.00 1379 2406
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 2.95 0.02 2.91 3.00 1.01 849 1650
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.38 0.01 0.37 0.39 1.00 7242 4307
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Model quantities and Statistics
Minimum and maximum betting times in the data:
## [1] 9## [1] 239Minimum and maximum in 11 draws from the posterior predictive distribution:
## [1] 3.781385 3.686012 3.687162 4.443887 4.337680 3.607308 4.754890 3.843820 3.554468 4.056925 4.064745## [1] 84.63636 81.90723 137.25560 86.65998 98.42180 105.45850 95.63466 98.44082 80.67333 135.69810
## [11] 108.53793Let’s take a look at posterior predicted mean and median
## [1] 18.75531 18.86243 18.93241 19.11055 18.90396 19.01423 19.08142 18.83936 18.95902 19.24667 18.90169## [1] 16.53884 16.54522 16.31937 16.67657 16.42843 16.63265 16.46356 16.42253 16.62015 17.00029 16.60487Calculate predicted mean and predicted median (based on lognormal distribution)
## # A tibble: 1 × 6
## mean .lower .upper .width .point .interval
## <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 20.6 19.7 21.6 0.95 mean hdi## # A tibble: 1 × 6
## median .lower .upper .width .point .interval
## <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 19.2 18.3 20.0 0.95 mean hdiShifted Lognormal Model
## Family: shifted_lognormal
## Links: mu = identity; sigma = identity; ndt = identity
## Formula: time ~ 1 + (1 | ppt_id)
## Data: bet_times_360 (Number of observations: 2573)
## Draws: 6 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup draws = 6000
##
## Group-Level Effects:
## ~ppt_id (Number of levels: 318)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sd(Intercept) 0.69 0.04 0.63 0.77 1.00 1065 2198
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept 2.13 0.05 2.04 2.22 1.00 884 1949
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.75 0.01 0.73 0.78 1.00 5997 5149
## ndt 8.71 0.04 8.63 8.78 1.00 6501 4903
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Model quantities and Statistics
Minimum and maximum betting times in the data:
## [1] 9## [1] 239Minimum and maximum in 11 draws from the posterior predictive distribution:
## [1] 8.856236 8.969384 8.868326 9.100594 8.953463 8.884594 8.890962 8.983805 9.062056 8.949329 9.052686## [1] 441.9064 176.6201 260.0013 243.7310 352.7175 199.0611 262.6505 187.9188 304.5261 185.9335 207.2581Let’s take a look at posterior predicted mean and median
## [1] 19.99761 19.95693 20.02788 20.34760 19.53222 19.83980 20.32771 20.24850 20.04705 20.18874 20.09058## [1] 15.15285 15.37562 15.58612 15.65409 15.21838 15.28248 15.34934 15.55657 15.32238 15.64694 15.14106Calculate predicted mean and predicted median (based on shifted-lognormal distribution)
## # A tibble: 1 × 6
## mean .lower .upper .width .point .interval
## <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 19.9 18.9 20.9 0.95 mean hdi## # A tibble: 1 × 6
## median .lower .upper .width .point .interval
## <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 17.2 16.4 17.9 0.95 mean hdiSession Info (R and Package Numbers)
## R version 4.1.2 (2021-11-01)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Ubuntu 20.04.3 LTS
##
## Matrix products: default
## BLAS: /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.9.0
## LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.9.0
##
## locale:
## [1] LC_CTYPE=en_GB.UTF-8 LC_NUMERIC=C LC_TIME=en_GB.UTF-8
## [4] LC_COLLATE=en_GB.UTF-8 LC_MONETARY=en_GB.UTF-8 LC_MESSAGES=en_GB.UTF-8
## [7] LC_PAPER=en_GB.UTF-8 LC_NAME=C LC_ADDRESS=C
## [10] LC_TELEPHONE=C LC_MEASUREMENT=en_GB.UTF-8 LC_IDENTIFICATION=C
##
## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
##
## other attached packages:
## [1] emmeans_1.7.2 binom_1.1-1 BayesFactor_0.9.12-4.3 Matrix_1.4-0
## [5] coda_0.19-4 tidybayes_3.0.2 brms_2.16.3 Rcpp_1.0.8
## [9] forcats_0.5.1 stringr_1.4.0 dplyr_1.0.8 purrr_0.3.4
## [13] readr_2.1.2 tidyr_1.2.0 tibble_3.1.6 ggplot2_3.3.5
## [17] tidyverse_1.3.1
##
## loaded via a namespace (and not attached):
## [1] readxl_1.3.1 backports_1.4.1 plyr_1.8.6 igraph_1.2.11
## [5] svUnit_1.0.6 splines_4.1.2 crosstalk_1.2.0 TH.data_1.1-0
## [9] rstantools_2.1.1 inline_0.3.19 digest_0.6.29 htmltools_0.5.2
## [13] rsconnect_0.8.25 fansi_1.0.2 magrittr_2.0.1 checkmate_2.0.0
## [17] tzdb_0.2.0 modelr_0.1.8 RcppParallel_5.1.5 matrixStats_0.61.0
## [21] vroom_1.5.7 xts_0.12.1 sandwich_3.0-1 prettyunits_1.1.1
## [25] colorspace_2.0-2 rvest_1.0.2 ggdist_3.1.1 haven_2.4.3
## [29] xfun_0.29 callr_3.7.0 crayon_1.5.0 jsonlite_1.7.3
## [33] lme4_1.1-28 survival_3.2-13 zoo_1.8-9 glue_1.6.1
## [37] gtable_0.3.0 MatrixModels_0.5-0 distributional_0.3.0 pkgbuild_1.3.1
## [41] rstan_2.21.3 abind_1.4-5 scales_1.1.1 mvtnorm_1.1-3
## [45] DBI_1.1.2 miniUI_0.1.1.1 xtable_1.8-4 HDInterval_0.2.2
## [49] diffobj_0.3.5 bit_4.0.4 stats4_4.1.2 StanHeaders_2.21.0-7
## [53] DT_0.20 htmlwidgets_1.5.4 httr_1.4.2 threejs_0.3.3
## [57] arrayhelpers_1.1-0 posterior_1.2.0 ellipsis_0.3.2 pkgconfig_2.0.3
## [61] loo_2.4.1 farver_2.1.0 sass_0.4.0 dbplyr_2.1.1
## [65] utf8_1.2.2 labeling_0.4.2 tidyselect_1.1.1 rlang_1.0.1
## [69] reshape2_1.4.4 later_1.3.0 munsell_0.5.0 cellranger_1.1.0
## [73] tools_4.1.2 cli_3.2.0 generics_0.1.2 broom_0.7.12
## [77] ggridges_0.5.3 evaluate_0.14 fastmap_1.1.0 yaml_2.2.2
## [81] bit64_4.0.5 processx_3.5.2 knitr_1.37 fs_1.5.2
## [85] pbapply_1.5-0 nlme_3.1-155 mime_0.12 projpred_2.0.2
## [89] xml2_1.3.3 compiler_4.1.2 bayesplot_1.8.1 shinythemes_1.2.0
## [93] rstudioapi_0.13 gamm4_0.2-6 reprex_2.0.1 bslib_0.3.1
## [97] stringi_1.7.6 highr_0.9 ps_1.6.0 Brobdingnag_1.2-7
## [101] lattice_0.20-45 nloptr_2.0.0 markdown_1.1 shinyjs_2.1.0
## [105] tensorA_0.36.2 vctrs_0.3.8 pillar_1.7.0 lifecycle_1.0.1
## [109] jquerylib_0.1.4 bridgesampling_1.1-2 estimability_1.3 httpuv_1.6.5
## [113] R6_2.5.1 promises_1.2.0.1 gridExtra_2.3 codetools_0.2-18
## [117] boot_1.3-28 colourpicker_1.1.1 MASS_7.3-55 gtools_3.9.2
## [121] assertthat_0.2.1 withr_2.4.3 shinystan_2.5.0 multcomp_1.4-18
## [125] mgcv_1.8-38 parallel_4.1.2 hms_1.1.1 grid_4.1.2
## [129] minqa_1.2.4 rmarkdown_2.11 shiny_1.7.1 lubridate_1.8.0
## [133] base64enc_0.1-3 dygraphs_1.1.1.6